Name: _________________________________________________ Period: ________ Score: ________

Unit 1: Solve Equations & Inequalities

Dates

for Sec 1

Dates

for Sec 1H

Section

#

Classwork: Objective

Classwork

Score 5 – 2 - 0

Homework

HW

On Time

HW

Effort Score:

Aug 20 & 21

Aug 20 & 21

Get to Know Each Other: How our class works,

Develop Class Norms, Working Together

Aug 22 & 23

Aug 22 & 23

1-1 Scavenger Hunt: Identify terms, coefficients, and constants.

HW 1-1

Aug 24 & 27

Aug 24 & 27

1-2 Why Do You Do What You Do? Solve equations and give mathematical justifications for each step in the process.

HW 1-2

Aug 28 & 29

Aug 28 & 29

1-3 Solve Multi-Step Equations

HW 1-3

Aug 30 & 31

Sep 10 & 11

1-4 Solve Absolute Value Equations

HW 1-4

Sep 4 & 5

Aug 30 & 31

1-5 Solve Literal Equations: Solve equations for a specified variable. QUIZ NEXT TIME

HW 1-5

Sep 6 & 7

Sep 4 & 5

1-6 Graph Compound Inequalities

HW 1-6

Sep 10 & 11

Sep 6 & 7

1-7 Solve and Graph Linear Inequalities

HW 1-7

Sep 12 & 13

Sep 10 & 11

1-8 Solve Absolute Value Inequalities

HW 1-8

Sep 14 - 19

Sep 12 - 17

Begin Unit 2

Sep 20 & 21

Sep 18 & 19

UNIT 1 TEST

4-Extra 3-Complete

2-Did What You Could

1-Started It

Equations Inequalities

Classwork 1-1: Terms, Coefficients, Expressions Welcome to world of math! During your stay here you will need to be familiar with some

vocabulary words. These words should not be new to you, but just in case, we will do a brief

review.

Term

Given: Example: Number of Terms:

4x2y

3x + 5

3x2 + 5x - 2 x

-7

4x2y

3x & 5

3x2 & 5x & -2 x

-7

1

2

3

1

1 Now in your own words define Term:

In the table above, label the algebraic expressions and the numeric expressions.

When we want to think of the number of terms the expression has, we call the expression a polynomial. The most common polynomials are monomial, binomial, and trinomial. Label the

above polynomials.

Coefficients

Given: Example: Counterexample:

4x2y

3x + 5

3x2 + 5x -2 x

7

4

3

3 & 5

1

0

2

3x, 5

-2

0

7 Now in your own words define Coefficient:

Score:_____/5__

Constant

Given: Example: Counterexample:

4x2y

3x + 5

3x2 + 5x - 2 x

-7

0

5

-2

0

-7

4, 2

3, 3x

3, 5

1, x

0 Now in your own words define Constant:

Terms: ___________________________ Terms: ___________________________

Variables: __________________ Variables: __________________

Coefficients: _______________ Coefficients: ______________

Constants: ________________ Constants: ________________

Can you combine any of the terms in the problems above?

Terms that can be combined are called: _________________________.

When you combine like terms you are simplifying the algebraic expression.

Simplify both of the above expressions:

_____________________________ ___________________________

Now how many terms does each expression have? _____ , _____

What is an Equation?

An equation says the two things are . . It must have an equal sign. . .

An Inequality is a mathematical sentence that contains the symbols: <, >, , or .

means _______________ means _________________

means _____________________ means _______________________

Scavenger Hunt

Your task is to find all of the following as if you were on a scavenger hunt. You need to work

with one or two other people and you need to work together as a team.

Write down the information that you find throughout the classroom.

1. An equation with a constant of 4 ________________________________________

2. A binomial _________________________________________________________

3. An equation that only has variables _______________________________________

4. An expression that has a coefficient of 5 _________________________________

5. An equation with a coefficient of 4

3 _____________________________________

6. A trinomial ________________________________________________________

7. An equation that has a trinomial equal to a variable ___________________________

8. An equation with a coefficient of 3 ________________________________________

9. A term with a coefficient of ½ ___________________________________________

10. A term that has an exponent of 3 ________________________________________

11. A monomial _________________________________________________________

12. An equation that finds the area of a triangle ________________________________

13. A linear equation _____________________________________________________

14. A nonlinear equation ___________________________________________________

Classwork 1-2: Why You Do What You Do

Solving equations has been part of your math experience for a long time. Some of the

steps you do in your head, while others are written down. Today you are going to look at the

reasons behind ALL the steps you do when solving equations.

1. Sarah has some money. For her birthday her grandma doubles the money she has. Then

Sarah spends $3 for the school dance. She now has $7. The following is an equation that

represents this situation. We want to find out how much money she had to start with, so we

solve the equation. Give a reason for each step of the solving process and what does it mean

according to the story.

2x – 3 = 7

2x – 3 + 3 = 7 + 3 _____________________________________________________________

2x + 0 = 7 + 3 __________________________________________________________

2x = 7 + 3 __________________________________________________________

2x = 10 ____________________________________________________________

½ ●2x = ½ ●10 _________________________________________________________

1●x = 5 ____________________________________________________________

x = 5 _____________________________________________________________

2. Now that you have given your reasons for each step, let’s look at the mathematical

properties that were used in each step. Compare these reasons to the reasons that you wrote

for # 1.

2x – 3 = 7

2x – 3 + 3 = 7 + 3 __________________________________________________________

2x + 0 = 7 + 3 _____________________________________________________________

2x = 7 + 3 ________________________________________________________________

2x = 10 __________________________________________________________________

½ ●2x = ½ ●10 _____________________________________________________________

1●x = 5 __________________________________________________________________

x = 5 ___________________________________________________________________

Score:_____/5

__

3. Given the following problem, give a property for each step.

2x – 5 = 4x + 7

2x – 5 + 5 = 4x + 7 + 5 _______________________________________________________

2x + 0 = 4x + 7 + 5 _______________________________________________________

2x = 4x + 7 + 5 ______________________________________________________

2x = 4x + 12 ________________________________________________________

2x - 4x = 4x – 4x + 12 _____________________________________________________

2x – 4x = 0 + 12 _________________________________________________________

2x - 4x = 12 ____________________________________________________________

-2x = 12 ____________________________________________________________

-½ ●-2x = -½ ●12 ________________________________________________________

1●x = -6 ____________________________________________________________

x = -6 ____________________________________________________________

4. List the mathematical property that is used for each step.

-2(n + 5) = -3n + 6

-2n – 10 = -3n + 6_______________________________________________________

-2n + 2n – 10 = -3n + 2n + 6____________________________________________________

0 – 10 = -3n + 2n + 6____________________________________________________

-10 = -3n + 2n + 6____________________________________________________

-10 = -1n + 6_______________________________________________________

-10 – 6 = -1n + 6 - 6_____________________________________________________

-10 – 6 = -1n + 0_______________________________________________________

-10 – 6 = -1n__________________________________________________________

-16 = -1n__________________________________________________________

-1●-16 = -1●-1n_______________________________________________________

16 = 1n__________________________________________________________

What are you noticing about solving equations?

Now using the ideas discussed in # 1-4, solve the following problems.

5. 5x + 2 = 17 6. 8k + 3k – 7 = 15

7. 6x – 25 = 7 – 2x 8. 2(n + 2) = 5n – 5

9. 8x – 7 = 3x – 7 + 5x 10. 4g + 7 = 5g – (1 + g)

Property Addition Multiplication Associative (𝑎 + 𝑏) + 𝑐 = 𝑎 + (𝑏 + 𝑐) (𝑎𝑏)𝑐 = 𝑎(𝑏𝑐)

Commutative 𝑎 + 𝑏 = 𝑏 + 𝑎 𝑎𝑏 = 𝑏𝑎

Identity 𝑎 + 0 = 𝑎 = 0 + 𝑎 𝑎 ∙ 1 = 𝑎 = 1 ∙ 𝑎

Inverse 𝑎 + (−𝑎) = 0 𝑎 ∙ 1

𝑎 = 1, 𝑖𝑓 𝑎 ≠ 0

Property of Equality

If 𝑎 = 𝑏, Then 𝑎 + 2 = 𝑏 + 2 If 𝑎 = 𝑏, Then 𝑎 ∗ 2 = 𝑎 ∗ 2

Distributive Property of Multiplication over Addition

𝑎(𝑏 + 𝑐) = 𝑎𝑏 + 𝑎𝑐

Simplify/Like Terms 3 + -7 = -4 or 3𝑥 + −7𝑥 = −4𝑥

Why You Do What You Do, Part 2

For each example, write the letter of the corresponding property in the blank provided. There are two new properties for you to consider.

Examples

______ 1. 6𝑦 + (−6𝑦) = 0

_______ 2. (𝑤𝑥)𝑦 = 𝑤(𝑥𝑦)

_______ 3. 4𝑥 + (𝑦 + 3𝑧) = (4𝑥 + 𝑦) + 3𝑧

_______ 4. (𝑤 + 𝑥)𝑦 = (𝑥 + 𝑤)𝑦

_______ 5. 8𝑥 + 3𝑥 = (8 + 3)𝑥

______ 6. If 4𝑥 + 1 = 8, then 4𝑥 + 1 + −1 = 8 + −1

_______ 7. (𝑤𝑥)𝑦 = 𝑦(𝑤𝑥)

_______ 8. (𝑤 + 𝑥)𝑦 = 𝑤𝑦 + 𝑥𝑦

_______ 9. 25 = 25(1)

_______ 10. −3𝑥 + 0 = −3𝑥

_______ 11. (𝑤𝑥)𝑦 = 𝑤(𝑥𝑦)

_______ 12. 4

5(

5

4) 𝑥 = 1𝑥

_______ 13. 𝑚 (1

𝑚) = 1, as long as 𝑚 ≠ 0.

_______ 14. 7 = 7

_______ 15. If 16 = 𝑥 + 2, then 𝑥 + 2 = 16

Properties

A. Addition Property of Equality B. Additive Identity C. Additive Inverse D. Associative Property of Addition E. Associative Property of Multiplication F. Commutative Property of Addition G. Commutative Property of Multiplication H. Distributive Property of Multiplication

Over Addition I. Multiplication Property of Equality J. Multiplicative Identity K. Multiplicative Inverse L. Reflexive property

M. Symmetric property

17. Explain the reflexive property in your own words, then give two additional examples.

18. Explain the symmetric property in your own words, then give two additional examples.

Classwork 1-3: Solving Multi-Step Equations

In your last classwork you learned that there are reasons for every step that you use when

solving equations. Your goal is to find the value of the variable. In general, what is the basic

order for solving equations?

Solve each equation. Make sure to show your work.

1. 4x – 5(2x -3) = 2x – 5 2. 2(x2 + 7) – 9 = 27

Solve the following problems.

3. 1.5x - 2.4 = 1.1x -1.6 4. 2

3𝑥 −

1

2=

3

4𝑥 −

2

3

When the equations have fractions or decimals, you can change them to whole numbers or you

can solve them in the same way as # 1, and #2.

5. In problem # 3, what would you multiply by to make all the numbers whole numbers?

What about # 4?

How do you choose what number to multiply by?

a. 1.5x - 2.4 = 1.1x -1.6 b. 2

3𝑥 −

1

2=

3

4𝑥 −

2

3

Score:_____/5__

Solve the following problems.

6. 2(x - 4) +6 = 2x – 2 7. 3x – 5 + 6 = 9x – 6x + 2

8. -4x + 5 - 2x = 10 - 3x – 5 9. 2𝑥−4

5=

4−2𝑥

3

10. How do you know if a problem has no solution, infinite solutions, or the variable has a value

of 0?

Write an equation for each problem below. Then solve the equation and find the length of

each side.

11. The perimeter of the triangle is the same as the perimeter of the rectangle.

12. The areas of both rectangles are the same.

4x+5

3x+12x

3x+2

x+7

30

x8

x+3

Classwork 1-4: Solving Absolute Value Equations

The absolute value of a number is its distance from zero on the number line without regard for direction. The symbol for

absolute value is |𝒙|, where x represents the number whose absolute value will be determined.

Example #1 |5| = 5, since 5 is a distance of 5 units from zero. |−5| = 5, since −5 is a distance of 5 units from zero.

Brian’s school has a spring carnival each year. One carnival game called the Absolute Aim involves the display of a number line, like the one shown. Here are the rules of Absolute Aim: • A contestant draws a card from a deck of five cards numbered 1, 2, 3, 4, and 5. • If the contestant can throw a dart the exact distance from zero that matches the drawn card’s value, then the

contestant wins a prize. 1. Brian draws a card with the number 4. Place dot(s) on the number line to represent the possible points the dart could

land on for Brian to win a prize.

a. Write an equation to show that Brian’s dart must be 4 units from zero on the number line. Let x represent the number associated with the location of Brian’s dart on the number line. b. List all possible solutions to your equation. 2. Jack draws a card with the number 3. Write an equation that shows Jack’s dart must be 3 units away from zero on the

number line. 3. The rules of the game have changed. The dart must now be the distance represented on the card from

+1 rather than

0.

a. Brian still has his card with the number 4. Place dot(s) on the number line to represent the possible points the dart could land to win a prize under the new rules.

b. Modify your equation from Question 1, part (a), to reflect the new rules.

Score:_____/5__

Distance is 5 units

●

Distance is 5 units

●

Algebraically, we can write the definition of |𝒙| in the following form.

|𝒙| = { 𝒙, 𝑖𝑓 𝑥 ≥ 0 −𝒙, 𝑖𝑓 𝑥 < 0

(Remember: −𝑥 is read as the opposite of x and can also be written as −1𝑥.) Let’s say that x is really 7y+8. Then |x| or |7y+8| is written as two equations: 7y+8 and –(7y+8).

If we wrote the equation for Brian’s dart throw as |𝑥 − 1| = 4, we can the absolute value equation into two separate equations and then solve the two equations.

First case: When 𝑥 − 1 ≥ 0

then |𝑥 − 1| = 𝑥 − 1.

4. Solve our new equation, 𝑥 − 1 = 4

Second case: When 𝑥 − 1 < 0

then |𝑥 − 1| = −(𝑥 − 1) or |𝑥 − 1| = −1(𝑥 − 1)

5. Solve our new equation, −1(𝑥 − 1) = 4

6. Verify that these two solutions make |𝑥 − 1| = 4 true. Rewrite each absolute value equation as two separate equations, and then solve both equations. When solving , remember that absolute value is a grouping symbol (parentheses). Verify your solutions.

7. |𝑥 + 7| = 3

8. 2|𝑥 − 9| = 12

9. |3𝑥 + 7| − 2 = 8

Classwork 1-5: Solve Literal Equations

Equations with several variables (letters) are called literal equations. Your job, usually, will be to

solve the equation for one of the variables. The letters that do not represent your desired variable

move to the other side of the equal sign so that the one variable you are solving for stands alone. This

means your “answer” is another equation. Solving literal equations have the exact same rules as

solving equations.

Equation Literal Equation

1 a) 2x – 5 = 11 b) Solve for b: 2b – 9 = d

2 a) 𝑥

−8 = 11 b) Solve for y:

𝑦

3 = h

3 a) 14 = 2x + 26 b) Solve for v: 3d = 7v + 5

4 a) 3(x – 4) = 12 b) Solve for p: 5(4x + p) = w

Formulas can be manipulated through the process of solving literal equations.

5. Write down 2-4 formulas that you know from math or science. Write what each variable

represents and what you are finding when you work out the formula.

Formula What does it mean?

______________________ __________________________________________

______________________ ____________________________________________

______________________ ______________________________________________

______________________ ______________________________________________

Score:_____/5__

Using your formulas from #5, let’s solve for a different variable:

Let’s practice some more:

6. Solve for b: A = 1

2bh (Area of a triangle)

7. Solve for r: C = 2πr (Circumference of a circle)

8. Solve for w: P = 2L + 2W (Perimeter of a rectangle)

9. Solve for C: F = 9

5C + 32 (Temperature conversions)

Given the following formulas solve for the indicated variable.

10. 3x + 2y = 12 for y 11. ½y + 2x = 4 for y

12. A = ½ (b1 + b2) for b2 13. A = πr2 for r

14. The formula below is for surface area of cylinders.

2

. . 2 2S A r rh

You are bottling a new energy drink. Because of cost, the can needs to be made of no more than 50

square inches of aluminum.

a. If the radius is 2 inches, what is the height?

b. Now solve the above equation for h:

You’re not sure that’s the best height of the can and want to look at other possibilities.

radius (inches) height (inches)

2.5

1.5

1.25

1

c. Which method is the most efficient in this example?

Classwork 1-6: Compound Inequalities

PART 1 – REVIEW An inequality is a mathematical sentence that relates two unequal quantities. Inequality symbols are placed between the two expressions.

1. What is the inequality symbol for:

“is less than” “is greater than”

“is less than or equal to” “is greater than or equal to” 2. Graph 𝑥 > 1 Graph 𝑥 ≥ 1

3. When do you use the open circle on the graph of an inequality?

4. When do you use the closed circle on the graph of an inequality?

PART 2 – COMPOUND INEQUALITIES

OR Statements

5. Jon will eat pizza or a hamburger for lunch.

If Jon eats tacos for lunch, is the above statement true or false?

If Jon eats pizza for lunch, is the statement true?

If Jon eats a hamburger for lunch, is it true?

6. 𝑥 ≤ 1 𝑜𝑟 𝑥 > 5 Graph all the points on the number line that make the compound statement true.

List 3 numbers that make it true (solutions of

the inequality).

7. a. Write an inequality for all the numbers less than -2 OR greater than or equal to 3.

b. Graph the compound inequality.

8. The city is having a carnival. They announce at the high school the carnival is for everyone 15 and

older. They announce at Willowcreek the carnival is for everyone 12 and older.

a. Write two inequalities to match the two statements.

b. Graph the inequalities on the same number line.

9. Graph the solutions to 𝟕 ≤ 𝒓 𝒐𝒓 𝒓 > −𝟑

Score:_____/5__

10. Kasey says she loves all numbers less than or equal to 10. She says she also loves all numbers

greater than 2.

a. Write two inequalities to match the two statements.

b. Graph the inequalities on the same number line.

11. Graph the solutions to 𝒌 > −𝟐 𝒐𝒓 𝒌 < 𝟑.

AND Statements

12. You have to be at least 45 inches and less than 86 inches tall to ride the rollercoaster.

a. Write the statement as two separate inequalities.

b. If you were 43” tall, could you get on the ride? c. If you were 87” tall, could you get on the ride?

d. If you were 60’ tall, could you get on the ride? e. Graph the inequalities on the same number line.

f. You can write the two inequalities in a compacted form.

13. a. Graph the solutions to 𝑥 > 2 𝑎𝑛𝑑 𝑥 ≤ 7 on the same number line.

b. Write the inequalities in compacted form.

14. “And” compound inequalities are often called the “intersection” of 2 inequalities. Why do you think

they use that term?

15. What is the difference between 𝒙 ≥ −𝟓 𝒐𝒓 𝒙 < 𝟖 and the compound inequality

𝒙 ≥ −𝟓 𝒂𝒏𝒅 𝒙 < 𝟖? Show their solutions on number lines.

Classwork 1-7: Solving Inequalities

For each instruction in 1 – 6, start with the inequality, 𝟓 > −𝟏𝟓 . Write the new inequality. Tell whether performing the operation will give you a true inequality.

1. Add 23 to both sides

2. Subtract 35 from both sides

3. Multiply both sides by 14.

4. Multiply both sides by -6.

5. Divide both sides by 5.

6. Divide both sides by -3.

*RULE* In your own words, write the rules for solving inequalities.

7. Charlotte has $800 in her savings account she uses for unexpected expenses. She finds that she’s spending

$50 each week. The equation 800 50x y represents the amount of money in her savings account, where

x represents the number of weeks. Write and solve an inequality to answer each question.

a. For what weeks will Charlotte have at least

$500?

b. For what weeks will Charlotte have no more

than $200?

c. For what weeks will Charlotte have less than

$0?

d. For what weeks will she have at least $400

or less than $-200?

Score:_____/5__

8. Leon plays on the varsity basketball team. So far this season he has scored a total of 52 points. He scores an

average of 13 points per game. The equation 13 52x y represents the total number of points Leon will

score this season, where x represents the number of games. Write and solve an inequality to answer each

question.

a. How many more games must Leon play

in order to score at least 117 points?

b. How many more games must Leon play in order to score fewer than 182 points?

c. How many more games must Leon play in order to score more than 143 points?

d. How many more games must Leon play in order to score at least 100 points?

Solve the inequality and then graph the solution on a number line. 9. 12 < 7𝑥 − 2

10. −2𝑥 + 12 > 32

11. −16𝑥 − 12 ≥ −2(8𝑥 + 7)

12. −17 − 4𝑥 ≤ 31

For problems like #13, rewrite the inequality as two inequalities using the conjunction “and” before

solving. Solve each compound inequality and then graph the solution on a number line.

13. 5𝑥 + 3 > 23 𝑜𝑟 − 2𝑥 > 4 Solution: ______________________

14. 6 < 𝑥 − 6 ≤ 9 Solution: ______________________

15. −6 < −3𝑥 < 18 Solution: ______________________

16. 6 + 3𝑥 > 15 𝑜𝑟 𝑥 − 4 < −5 Solution: ______________________

17. 2𝑥 − 1 < 5𝑥 + 2 𝑎𝑛𝑑 5𝑥 + 6 ≤ 3𝑥 + 16 Solution: ______________________

Classwork 1-8: Solving Absolute Value Inequalities

To solve Absolute Inequalities, you will use the exact same rules that you learned while solving inequalities (1 -6) and while solving absolute value equations (1-7), but are now used on the same problem. You will see these rules as you play Absolute Aim with Brian and Zach again. 1. Brian is still playing the game Absolute Aim and draws a card that states he needs to throw his dart AT LEAST 6

units away in order to win. If he is at zero, show on the number line the possible points the dart could land on for Brian to win a prize.

a. Write an inequality to show that Brian’s dart must be 6 units from zero on the number line. Let x represent the number associated with the location of Brian’s dart on the number line. b. Write the solutions to your inequality. 2. Jack draws a card that states his dart needs to land NO MORE THAN 7 units away from 0.

a. Write an inequality that shows where Jack’s dart can land.

b. Write the solutions to your inequality.

3. a. Brian now draws a card that states he is to stand on 3 and throw his dart AT LEAST 6 units away in order to win.

Show on the number line where his dart can land. b. Modify your inequality from Question 1, part (a), to reflect the new rules.

c. Using this inequality, the rules you know to solve absolute value equations and the rules for inequalities, solve this absolute value inequality for Brian.

4. What are the rules to follow when solving an absolute value inequality?

Score:_____/5__

Rewrite each absolute value inequality as two separate inequalities, and then solve both. When solving, remember that absolute value is a grouping symbol (parentheses). Verify your solutions and graph them.

5. |𝑥 + 7| ≥ 3

6. 2|𝑥 − 9| < 12

7. |3𝑥 + 7| − 2 > 8

Honors Supplement Problems

Lesson 1-3

13. A large pizza at Palanzio’s Pizzeria costs $6.80 plus $0.90 for each topping. The cost of a large cheese

pizza at Guido’s Pizza is $7.30 plus $0.65 for each topping. How many toppings need to be added to a large

cheese pizza from Palanzio’s Pizzeria and Guido’s Pizza in order for the pizzas to cost the same, not including

tax?